For a compact space Y, we view C(Y × S1) as the crossed product C(Y ) ⋊ <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathbb{Z}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbb{Z}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation> acting trivially. This allows us to study Rieffel projections in M2(C(Y × S1): we characterize them and compute their image under the projection ∂0 : K0(C(Y × S1)) → K1(C(Y )). We provide a new Rieffel projection in M2(C(<InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathbb{T}}^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>)): in contrast with Loring’s projection [<CitationRef CitationID="CR16">16</CitationRef>], which involves nonalgebraic functions, ours involves only trigonometric polynomials plus the square root of 2−e2πiθ−e−2πiθ. We give applications of this projection, for example, explicit generators for the K-theory of C(<InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathbb{T}}^{3}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>). Finally, we prove that if a Banach algebra completion <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal{B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\mathbb{C}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>[<InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\mathbb{Z}}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>] is continuously contained in C(<InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\mathbb{T}}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>) and such that the Fourier series of (2 − e2πiθj − e−2πiθj)1/2 (j = 1, . . . , n) converges in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal{B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation>, then the inclusion <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal{B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation> ↪ C(<InlineEquation ID="IEq11"> <EquationSource Format="TEX">${\mathbb{T}}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>) induces isomorphisms in K-theory.

Rieffel projections and 2-by-2 matrices

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2026年7月6日

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Olivier Isely,
Alain Valette

摘要

For a compact space Y, we view C(Y × S¹) as the crossed product C(Y ) ⋊ ${\mathbb{Z}}$ $Z$ , with ${\mathbb{Z}}$ $Z$ acting trivially. This allows us to study Rieffel projections in M₂(C(Y × S¹): we characterize them and compute their image under the projection ∂₀ : K₀(C(Y × S¹)) → K₁(C(Y )). We provide a new Rieffel projection in M₂(C( ${\mathbb{T}}^{2}$ $T^{2}$ )): in contrast with Loring’s projection [16], which involves nonalgebraic functions, ours involves only trigonometric polynomials plus the square root of 2−e^2πiθ−e^−2πiθ. We give applications of this projection, for example, explicit generators for the K-theory of C( ${\mathbb{T}}^{3}$ $T^{3}$ ). Finally, we prove that if a Banach algebra completion $\mathcal{B}$ $B$ of ${\mathbb{C}}$ $C$ [ ${\mathbb{Z}}^{n}$ $Z^{n}$ ] is continuously contained in C( ${\mathbb{T}}^{n}$ $T^{n}$ ) and such that the Fourier series of (2 − e^2πiθj − e^−2πiθj)^1/2 (j = 1, . . . , n) converges in $\mathcal{B}$ $B$ , then the inclusion $\mathcal{B}$ $B$ ↪ C( ${\mathbb{T}}^{n}$ $T^{n}$ ) induces isomorphisms in K-theory.