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14

I know two-ish answers to this question. Representation-theoretic: The category of representations of a group has both tensor products and duals, but the category of representations of a general algebra generally has neither (or at least there is no obvious way to define them). Since the category of representations of a group $G$ is equivalent to the ...


11

One application of this is that topological K-theory (i.e., the K-theory of the exact category of vector bundles on the space) is the same thing as the "algebraic" $K_0$ of the ring of continuous functions (i.e., the K-theory of the exact category of finitely generated projective modules over that ring). So topological K-theory is a "special case" of ...


8

Roughly, the answer will be that closed C*-subalgebras will correspond to quotient spaces (via pull-back of functions). In your example, the quotient map is one which identifies the two points into a single point. I haven't thought through, though, whether this is a completely correct statement as it stands, or whether one has to add additional caveats. ...


6

You could perhaps do worse than consulting $\S 6.4$ of my commutative algebra notes: "Applications of Swan's Theorem." (You could definitely do better: see below.) The first application I give is to show that the ring of real-valued continuous functions on $[0,1]$ is a connected ring in which each finitely generated projective module is free but for which ...


6

In a recent paper, I adapted some of the terminology of Kochen and Specker's original paper to a more ring-theoretic context. I would refer to your first type of morphism as a morphism of partial $\mathbb{Z}$-algebras from $R$ to $S$, or even better as a morphism of partial rings. The point is that every ring has the underlying structure of a partial ring ...


6

Before Drinfeld's work in the 1980s there was only marginal interest by mathematicians in general (noncommutative, noncocommutative) Hopf algebras, so it could be difficult to honestly interest students in the Hopf algebra axioms without mentioning quantum groups. On this view the place to start is Drinfeld's ICM lecture and papers, or books on quantum ...


6

I think Qiaochu's answer is the best, but I thought that two topological reasons should be given. The first is that the Steenrod squares form a Hopf algebra, and their structure is one of the most important aspects of homology theory. The second is the diagrammatic interpretation of the axioms. Letting $\lambda$ denote the multiplication operator, and $Y$ ...


6

They're (co)group-objects, for one. That's always fun!


5

You can do this for an arbitrary ring (with or without unit). Jacobson's original article can be found here (JSTOR, needs a university subscription). I cannot do better than to simply quote C. Chevalley's Math Review (MathSciNet, needs a university subscription): A (two-sided) ideal $\frak J$ in a ring $\frak A$ is called primitive if $0$ is the only ...


5

I'm afraid I'm rather late to the party, but let me throw out a few thoughts, in the hope that something will be of use to someone. You probably know everything under 1. and 2., so if you want the punchline, do forgive the tl;dr and just skip ahead to 3. To be absolutely clear about the state of the art, Connes's theorem actually tells you the following: ...


5

If I'm reading your question correctly (you may be asking several related things and I am not sure exactly what is a question and what is a discussion), the analogous theorem in the Banach algebra setting is the commutative Gelfand-Naimark theorem. The geometric thing is compact Hausdorff spaces and the algebraic thing is commutative unital C*-algebras. The ...


5

Take $A = L^{\infty}[0,1]$ with pointwise multiplication and let $B = C[0,1]$ be the closed $C^{\ast}$-subalgebra of continuous functions. The $\ast$-subalgebra $D \subset A$ consisting of the simple functions (finite linear combinations of characteristic functions of measurable sets) is dense in $L^{\infty}[0,1]$ but $D \cap B = \{\text{constant ...


4

Let $A=M_2(\Bbb Z)$ be $2\times 2$ matrices over $\Bbb Z$ and let $x=2I$ be twice the identity.


3

One instance where an operator theory problem was solved via "noncommutative topology" (actually it is the introduction of that viewpoint into operator theory) is the Brown-Douglas-Fillmore theory for classifying essentially normal operators. They found that, in addition to the essential spectrum, the Fredholm index was the key ingredient for classification ...


3

I would recommend that you learn some Operator Algbras first, from say Murphy's Operator Algebras and Operator Theory. Then you can learn some K-theory from Rordam/Larsen/Laustsen's book Introduction to K-theory for Operator Algebras. After that, you should be knowledgeable enough to find your own way around.


3

It comes from composition of isomorphisms. One version of the "algebra of functions" on, say, a finite groupoid $G$ is its groupoid algebra $\mathbb{C}[G]$, which is a direct generalization of the group algebra: take the free vector space on the morphisms in $G$ with multiplication given by composition (or $0$ if there is no composition). If $G$ is an ...


3

After many days of consistent effort, finally I could find a complete article that speaks about proving the various corollaries and theorems of Gauss-Bonnet , even though the article starts with a preliminary version of the proof ( considering Riemann metrics and the Euler forms, it does have a different versions of the proof, and the whole article is ...


3

One way that Hopf algebras come up is as the algebra of (real or complex) functions on a topological group. The multiplication is commutative since it is just pointwise multiplication of functions. However, in non-commutative geometry you want to replace the algebra of functions on a space with a non-commutative algebra, giving a non-commutative Hopf ...


3

Let $\text{MinSpec}(A) \subset \text{Spec}(A)$ be the subset of minimal primes. I claim that $\text{MinSpec}(A) \cap W$ is open and closed whenever $W$ is a quasi-compact open of $\text{Spec}(A)$. This follows immediately from Lemma Tag 00EV (which has a purely algebraic proof). Hence $\text{MinSpec}(A)$ has a base for its topology consisting of closed and ...


3

The term you want is "distributes over," not "commutes with." (Whatever "$A$ commutes with $B$" means it should be symmetric in $A$ and $B$, which the condition you want is not.) Such monoidal categories are called distributive. Examples include any closed monoidal category with coproducts because in this case $A \otimes (-)$ has a right adjoint and hence ...


3

1) What is the physical interpretation of q-deformation? It doesn't look like it is the same as going from classical mechanics to quantum mechanics. I have seen q-deformation in the case of quantum groups, though nothing was said about the physical interpretation. A relation between the two notions is If $[x,y]=h$ (a scalar or central element) ...


3

An orientable smooth manifold $X$ admits a $\text{Spin}^c$ structure iff its second Stiefel-Whitney class $w_2 \in H^2(X, \mathbb{F}_2)$ is the reduction of a class $c_1 \in H^2(X, \mathbb{Z})$. This condition is equivalent to the condition that the third integral Stiefel-Whitney class $W_3 = \beta w_2 \in H^3(X, \mathbb{Z})$ vanishes, and I guess this is ...


3

Standard references for basic C*-algebra theory include: C*-Algebras by Example by Kenneth R. Davidson C*-Algebras and Operator Theory by Gerard J. Murphy An Introduction to K-Theory for C*-Algebras by M. Rørdam, F. Larsen, N. Laustsen In general, it seems safe to state that a solid knowledge of algebraic topology and functional analysis is useful for ...


2

There's nothing really systematic, but there is a fair bit of literature already on what you might call discrete noncommutative Riemannian geometry à la Connes, dealing with noncommutative spectral triples that are discrete in some sense. On the one hand, there's been a great deal of work on so called "finite" or "discrete" spectral triples, spectral ...


2

A survey: http://www.mast.queensu.ca/~speicher/survey.html also worth a read: http://www.ams.org/notices/200405/comm-nas.pdf


2

The Leroy P. Steele Prize is awarded for expository mathematical works. I share your appreciation of aesthetic value in math literature (which many authors evidently see as unnecessary).


2

Yes and yes. To fix notation, what we're dealing with is the commutative spectral triple $(C^\infty(X),L^2(X,\wedge T^\ast X),d+d^\ast)$, where $(X,g)$ is a compact oriented Riemannian manifold. Recall, in particular, that $\wedge T^\ast X$ is a Hermitian vector bundle with Hermitian metric $(\cdot,\cdot)$ induced by the Riemannian metric $g$, so that the ...


2

This is in Cyclic Homology by Loday, theorem 1.4.14: Theorem: Let $I$ be a H-unital $k$-algebra. Then $M_r(I)$ (matrices of size $r$) is $H$-unital for all $r$ (including $r = > \infty$), and the maps (resp. induced by the trace and the inclusion): $$\begin{align} \mathrm{tr}_* & : HH_*(M_r(I), M_r(M)) \to HH_*(I,M) \\ \mathrm{inc}_* & : ...


2

Recall that any $f \in C^\infty(\mathbb{T}^n)$ can be uniquely written as a convergent Fourier series $$ f = \sum_{\mathbb{k} \in \mathbb{Z}^n} f_{\mathbb{k}} U_{\mathbb{k}}, \quad f_{\mathbb{k}} := \int_{\mathbb{T}^n} e^{-2\pi i \langle \mathbb{k},t \rangle}f(t)\,dt, $$ where for each $\mathbb{k} \in \mathbb{Z}^n$, $$ \forall t \in \mathbb{T}^n, \quad ...


2

I cannot speak about the cohomology thing because I would have to go back and look at the definitions. But it is trivially true that every von Neumann algebra has hyperfinite subalgebras. In fact, every von Neumann algebra has finite-dimensional unital subalgebras. You can start with $\mathbb C\,1$, for instance. If the von Neumann algebra is nontrivial it ...



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