Questions tagged [kk-theory]

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Functional Calculus for the Dolbeault Operator over Hilbert C*-modules

Let $X$ be a complex manifold, you can assume it's compact, if necessary. We have the Dolbeault complex $$0 \rightarrow \mathcal{A}^{0,0} \xrightarrow{\bar{\partial}} \mathcal{A}^{0,1} \xrightarrow{\...
5
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346 views

KK-witnesses of Gysin maps between differentiable stacks

In 1982 Alain Connes gave the construction of a KK-element $f! \in KK(C(X), C(Y))$ that "witnesses" the fiber integration/Gysin/Umkehr/wrong-way map on topological $K$-theory along a K-orientable map ...
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139 views

Solution without using any k-theory tools

Let $A$ be the UHF-algebra of type $2^{\infty}$. Suppose that $p$ and $q$ are two projections in $A$ and $\tau(p) = \tau(q)$, where $\tau$ is the unique normalized trace. Then there is a partail ...
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213 views

Künneth formula for $C^*$ algebras, equivalent condition for full generality

I'm crrently reading the paper about the Künneth-theorem for $C^*$-algebras: http://msp.org/pjm/1982/98-2/pjm-v98-n2-p15-s.pdf and I'm trying to understand remark 4.9. I henceforth asumme that $A$ is ...
3
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104 views

Another way for defining $K_1$ group for a C*-algebra

Thank you for answering my question. I have another question about the $K_1$ group. As you may know, some books define the $K_1$ group like below: Also, it defines the $K_0$ group for an arbitrary C*-...
3
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219 views

The Green-Julg Theorem

I am currently trying to understand the general Green-Julg theorem, where $G$ is a compact group, $A$ and $B$ are $G$-$C^*$-algebras, and where $G$ acts trivially on $A$. The Green-Julg theorem states ...
2
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97 views

Representation of $C^{*} (S_{\infty})$

I was wondering what is the group $C^{*}$-algebra of infinite symmetric group? Mainly, I was trying to calculate the k-theory of $C^{*}$-algebra of infinite symmetric group and I found K-Theory of $C^{...
2
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75 views

Relative de Rham Cohomology groups of k-algebra

Let $A$ be a commutative unital $k$-algebra. Then we have de Rham complex given as: $C_{\ast}(A)$ : $ 0 \rightarrow A \rightarrow \Omega_{A \lvert k}^{1} \rightarrow \Omega_{A \lvert k}^{2} \...
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242 views

Separable $\sigma$-unital sub-$C^*$-Algebras

Let $A$ be a $\mathbb{Z}_2$-graded $C^*$-Algebra. Then we can take the direct limit $$ colim_{A_\sigma} KK_*(\mathbb{C}, A_\sigma) $$ over all $\sigma$-unital graded $C^*$-sub-algebras $A_\sigma \...
1
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A map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov

Let $ G $ be a locally compact group. Let $ H $ and $ K $ be two normal subgroups of $ G $. In order to construct a map, $$ \psi \ : \ \ F(G/H,G/K) \to F(G/K,G/H) $$ where, $$ F(G/H,G/K) = KK^{G/H} ( ...
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96 views

Kasparov's descent homomorphism for higher KK groups

I am currently trying to understand equivariant $KK$-theory. I think I roughly get the idea of Kasparov's descent homomorphism $$KK^G(A,B) \rightarrow KK(A \rtimes G,B \rtimes G).$$ but what still ...
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55 views

When is a cycle in $KK^G(A,A)$ with zero operator the identity cycle?

Given a cycle of the form $(\pi,H,0)$ in $KK^G(A,A)$, when is it equivalent to the identity cycle $1_A=(i_A,A,0)$? The operator $T=0$, and $\pi:A \rightarrow L(H)$ may be injective. Any criterions ...
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71 views

Is $KK^G(\mathbb{C}^n,B)$ countably additive in $B$ and countable?

Let $G$ be a finite discrete groupoid, $A=\mathbb{C}^n$ a finite dimensional, commutative $C^*$-algebra and assume we have given a $G$-action on $A$. Note that the action of $G$ on $\mathbb{C}^n=C_0(\{...
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279 views

Lifting triangles in K-theory to KL-groups

Let $X$ and $Y$ be finite simplicial complexes (or $CW$-complexes) so that $Y\subseteq X$. Let $s\colon C(X)\to C(Y)$ be the map given by restriction. In particular $K_{*}(C(X))$ and $K_{*}(C(Y))$ are ...